Conductance distributions in disordered quantum spin-Hall systems

We study numerically the charge conductance distributions of disordered quantum spin-Hall (QSH) systems using a quantum network model. We have found that the conductance distribution at the metal-QSH insulator transition is clearly different from that at the metal-ordinary insulator transition. Thus the critical conductance distribution is sensitive not only to the boundary condition but also to the presence of edge states in the adjacent insulating phase. We have also calculated the point-contact conductance. Even when the two-terminal conductance is approximately quantized, we find large fluctuations in the point-contact conductance. Furthermore, we have found a semicircular relation between the average of the point-contact conductance and its fluctuation.

Figure 1

(Color online) Schematics of the (a) $\mathit{s}$-type and (b) ${\mathit{s}}^{\prime }$-type nodes described by the scattering matrices ${\mathit{s}}_i$ and ${\mathit{s}}_i^{\prime }$, respectively. The arrow on the upper (under) side of a link indicates the direction of currents for up (down) spin.

Figure 2

(Color online) Schematics of the ${\mathbb{Z}}_2$ networks in a square geometry with RBC. The left and right figures correspond to the network model at $p\to 0$ and 1, respectively, while the middle figure represents the metallic phase at intermediate $p$. When $p$ is close to 0 or 1, all the bulk states are localized in plaquettes. In the QSH insulating phase $\left(p\to 1\right)$, however, two Kramers pairs of conducting channels run along the edges, which correspond to the helical edge state.

Figure 3

(Color online) Phase diagram (Ref. 12) of ${\mathbb{Z}}_2$ network model with RBC. Note that the phase boundaries between metallic and insulating phases are the same for networks with PBC. For PBC, the ordinary insulating phase replaces the QSH insulating phase. Circled letters indicate the points considered in the corresponding sections in Sec. 3. The actual values of parameters $\left(p,q\right)$ are as follows: $\mathbf{A}:\left(0.365,0.500\right)$, $\mathbf{B}:\left(0.039,0.146\right)$, ${\mathbf{B}}^{\prime }:\left(0.783,0.146\right)$, $\mathbf{C}:\left(0.188,0.309\right)$, $\mathbf{D}:\left(0.561,0.309\right)$, and ${\mathbf{D}}^{\prime }:\left(0.563,0.146\right)$.

Figure 4

(Color online) Conductance distributions $P\left(G\right)$ in the metallic phase $\mathbf{A}$ with RBC, for $L=64$, 128, and 256 (from left to right). Distributions keep Gaussian shapes (dotted line) for increasing system size.

Figure 5

(Color online) Averaged conductances as a function of system size in the metallic phase $\mathbf{A}$ with PBC (○) and RBC (◻). The solid lines represent the antilocalization formula, Eq. (8).

Figure 6

(Color online) Averaged conductance as a function of system size, for ordinary insulator $\mathbf{B}$ with PBC (○) and RBC (◻), and QSH insulator ${\mathbf{B}}^{\prime }$ (×).

Figure 7

(Color online) Conductance distributions $P\left(\text{ln}G\right)$ in ordinary insulating phase $\mathbf{B}$ with RBC for $L=256$ (—–) and with PBC for $L=128$ (- - -). For ordinary insulating phase, the distribution functions of $\text{ln}G$ are well fitted by Gaussian functions (dotted line) for any system size. For QSH insulating phase, distribution functions $P\left(G\right)$ asymptotically approach to the delta function $\delta \left(G-2\right)$.

Figure 8

(Color online) Conductance distributions at the M-OI transition point $\mathbf{C}$ calculated on the ${\mathbb{Z}}_2$ network model (—), S2NC model (○), and SU(2) model (×) for PBC and RBC [FBC for the SU(2) model]. More than ${10}^6$ independent random configurations have been realized to draw each curve.

Figure 9

(Color online) Conductance distributions at the M-QSHI transition with $L=256$. The distributions at the different points $\mathbf{D}$ (—–) and ${\mathbf{D}}^{\prime }$ (- - -) almost coincide with each other. ${10}^6$ samples are calculated for each critical point.

Figure 10

(Color online) Distribution functions of the largest $P\left(2{\tau }_1\right)$ (—–) and second-largest $P\left(2{\tau }_3\right)$ (- - -) transmission eigenvalues for M-OI transitions $\mathbf{C}$ with PBC and RBC. Dotted lines are the distribution functions of the conductance $P\left(G\right)$ in Fig. 8 for $G<2$ and for $G>2$, both of which are normalized to be 1. The latter is shifted by $-2$ along the horizontal axis to be compared with $P\left(2{\tau }_3\right)$.

Figure 11

(Color online) Distribution functions of the largest $P\left(2{\tau }_1\right)$ (—–) and second-largest $P\left(2{\tau }_3\right)$ (- - -) transmission eigenvalues for M-QSHI transition $\mathbf{D}$. Dotted lines are the distribution functions of the conductance $P\left(G\right)$ in Fig. 9 for $G<2$ and for $G>2$, both of which are normalized to be 1. The latter is shifted by $-2$ along the horizontal axis to be compared with $P\left(2{\tau }_3\right)$.

Figure 12

(Color online) Schematics of the ${\mathbb{Z}}_2$ networks with point contacts (a) on the edge, $y_{\text{p}}=1$, and (b) in the bulk region, $y_{\text{p}}=L/2$. Each point contact is connected with a link. PBC are imposed on the longitudinal direction.

Figure 13

(Color online) Current amplitude $\left({\left|c_{i\uparrow }\right|}^2+{\left|c_{i\downarrow }\right|}^2\right)$ of a ${\mathbb{Z}}_2$ network with $\left(p,q\right)=\left(0.750,0.146\right)$, for $L=80$. The contacts are attached at the upper edge.

Figure 14

(Color online) Distribution functions of the point-contact conductance at the upper edge, $y_{\text{p}}=1$ (—–), near the edge, $y_{\text{p}}=3$ (- - -), and at a distance on the order of the localization length, $y_{\text{p}}=19$ $(\cdots )$, with $\left(p,q\right)=\left(0.750,0.146\right)$, for $L=80$. ${10}^5$ samples are realized. Although the two-terminal conductance is quantized for corresponding parameters, large fluctuations appear for the point-contact conductance.

Figure 15

(Color online) (a) Averages and (b) fluctuations of edge ($◻:L=80$, $+:L=160$) and bulk ($○:L=80$, $\times :L=160$) point-contact conductances with $q=0.146$ as functions of $p$. The lines are guide to the eyes. The plateau of fluctuation of the bulk conductance corresponds to the metallic phase.

Figure 16

(Color online) Fluctuations as a function of the averaged point-contact conductance for edge ($◻:L=80$, $+:L=160$), $y_{\text{p}}=1$, and bulk ($○:L=80$, $\times :L=160$), $y_{\text{p}}=L/2$, with $q=0.146$. This shows a specific relation similar to a semicircle (—–) between the average and fluctuations. We have confirmed that the same relation is satisfied for different geometry including boundary conditions, other contact point $\left(y_{\text{p}}=3\right)$, or different spin-flip rate $\left(q=0.309\right)$. Statistical errors are smaller than symbols.

Figure 17

(Color online) Fluctuations as a function of the averaged point-contact conductance in the Chalker-Coddington model (IQH system). Plots for both edge ($◻:L=80$, $+:L=160$), $y_{\text{p}}=1$, and bulk ($○:L=80$, $\times :L=160$), $y_{\text{p}}=L/2$, agree with the semicircular relation (—–). The result for the single channel DMPK equation (Refs. 29, 30) (- - -) agrees with the semicircle only for small averaged conductance.